A practical guide to analytical method validation, including measurement uncertainty and accuracy profiles

Trends in Analytical Chemistry, Vol. 26, No. 3, 2007 Trends

A practical guide to analytical
method validation, including
measurement uncertainty and
accuracy profiles
´ ´
A. Gustavo Gonzalez, M. Angeles Herrador

The objective of analytical method validation is to ensure that every future also to evaluate those risks that can be
measurement in routine analysis will be close enough to the unknown true expressed by the measurement uncer-
value for the content of the analyte in the sample. Classical approaches to tainty associated with the result [2].
validation only check performance against reference values, but this does not Accuracy, according to the ISO 5725
reflect the needs of consumers. A holistic approach to validation also takes definition [3], comprises two components
into account the expected proportion of acceptable results lying inside – trueness and precision – but, instead of
predefined acceptability intervals. assessing these independently, it is possible
In this article, we give a detailed step-by-step guide to analytical method to assess accuracy in a global way
validation, considering the most relevant procedures for checking the quality according to the concept of acceptability
parameters of analytical methods. Using a holistic approach, we also explain limits and accuracy profiles [4–8]. Accu-
the estimation of measurement uncertainty and accuracy profiles, which we racy profiles and measurement uncer-
discuss in terms of accreditation requirements and predefined acceptability tainty are related topics, so either can be
limits. evaluated using the other. In a holistic
ª 2007 Elsevier Ltd. All rights reserved. sense, as Feinberg and Laurentie pointed
out [9], method validation, together with
Keywords: Accuracy profile; Analytical method validation; Measurement uncertainty;
uncertainty measurement or accuracy-
Performance characteristic
profile estimation, can provide a way to
Abbreviations: ANOVA, Analysis of variance; bETI, b-expectation tolerance interval; CS, check whether an analytical method is
Calibration standard; DTS, Draft technical specification; ELISA, Enzyme-linked correctly fit for the purpose of meeting
immunosorbent assay; FDA, Food and Drug Administration; ICH, International legal requirements. Fitness for purpose is
Conference on Harmonization; ICP, Inductively coupled plasma; ISO, International the extent to which the performance of a
Organization for Standardization; IUPAC, International Union of Pure and Applied method matches the criteria that have
Chemistry; LGC, Laboratory of Government Chemist; LOD, Limit of detection; LOQ, been agreed between the analyst and the
Limit of quantitation; RSD, Relative standard deviation; SAM, Standard-addition
end-user of the data or the consumer and
´
method; SB, System blank; SFSTP, Societe Francaise des Sciences et Techniques
¸
that describe their needs [10]. Classical
Pharmaceutiques; TR, Tolerated ratio; TYB, Total Youden blank; USP, United States
Pharmacopoeia; VAM, Valid analytical measurement; VS, Validation standard; YB,
approaches to validation consisted of
Youden blank. checking the conformity of a performance
measure to a reference value, but this does
1. Introduction not reflect the consumerÕs needs, men-
´
A. Gustavo Gonzalez*,
´
tioned above. By contrast, the holistic ap-
M. Angeles Herrador
Department of Analytical
The final goal of the validation of an proach to validation establishes the
Chemistry, University of Seville, analytical method is to ensure that every expected proportion of acceptable results
E-41012 Seville, Spain future measurement in routine analysis lying between predefined acceptability
will be close enough to the unknown true limits. Many excellent papers and guides
*
value for the content of the analyte in the have been written about the validation of
Corresponding author.
Tel.: +34 954557173;
sample. [1]. Accordingly, the objectives of analytical methods but no attention has
Fax: +34 954557168; validation are not simply to obtain esti- been paid to the holistic paradigm. We aim
E-mail: agonzale@us.es mates of trueness or bias and precision but to provide to the analyst with a practical

0165-9936/$ - see front matter ª 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.trac.2007.01.009 227

Trends Trends in Analytical Chemistry, Vol. 26, No. 3, 2007

guide to performing the validation of analytical methods applicability must be consistent with the ‘‘golden rules’’
using this holistic approach. for method validation proposed by Massart et al. [11],
namely:
(1) the analytical procedure has to be validated as a
2. Practical approach to global method validation whole, including sample treatments prior to analy-
sis;
For the sake of clarity, we have divided the content of the (2) the analytical procedure has to be validated cover-
guide into four sections that we will outline and explain, ing the full range of analyte concentrations specified
as follows and as shown in Fig. 1: in the method scope; and,
(1) applicability, fitness for purpose and acceptability (3) the analytical procedure has to be validated for each
limits; kind of matrix where it will be applied.
(2) specificity and selectivity; Fitness for purpose [10,12,13] is the extent to which
(3) calibration study, involving the goodness of the fit the method performance matches the agreed criteria or
of the calibration function and dynamic concentra- requirements. A laboratory must be capable of providing
tion range, sensitivity and detection and determina- results of the required quality. The agreed requirements
tion limits, as well as assessment for matrix effects; of an analytical method and the required quality of the
and, analytical result (i.e. its accuracy) refer to the fitness for
(4) accuracy study, involving trueness, precision and purpose of the analytical method. The accuracy can be
robustness as well as the estimation of measurement assessed in a global way, as indicated above, by using the
uncertainty and accuracy profiles concept of acceptability limit k [6–8]. Thus, analytical
result Z may differ from unknown ‘‘true value’’ T to an
2.1. Applicability, fitness for purpose and acceptability extent less than the acceptability limit:
limits jZ À Tj < k ð1Þ
The method applicability is a set of features that cover,
apart from the performance specifications, information Limit k depends on the goals of the analytical proce-
about the identity of analyte (e.g., nature and specia- dure: 1% for bulk materials; 5% for determination of
tion), concentration range covered, kind of matrix of the active ingredients in dosage forms; and, 15% in bio/
material considered for validation, the corresponding environmental analysis [8]. A procedure can be vali-
protocol (describing equipment, reagents, analytical dated if it is very likely that the requirement given by (1)
procedure, including calibration, as well as quality pro- is fulfilled, i.e.:
cedures and safety precautions) and the intended appli-
PðjZ À lj < kÞ P b ð2Þ
cation with its critical requirements [10]. The method
b being the probability that a future determination
falls inside the acceptability limits. It is possible to com-
pute the so-called ‘‘b-expectation tolerance interval’’
HOLISTIC VALIDATION APPROACH (bETI) (i.e. the interval of future results that meet
Equation (2)) by using the accuracy profiles that we will
describe later (in Section 2.4., devoted to accuracy study
Applicability, fitness for purpose and and measurement uncertainty). The use of acceptability
acceptability limits
limits together with accuracy profiles is an excellent way
to check the fitness for purpose of the validated method.
Selectivity and Specificity

Calibration Study 2.2. Specificity and selectivity
Selectivity is the degree to which a method can quantify
Goodness of the fit and linear range the analyte accurately in the presence of interferences
Sensitivity, LOD and LOQ under the stated conditions of the assay for the sample
Assessment of constant and
matrix being studied. As it is impracticable to consider
proportional bias every potential interference, it is advisable to study only
the worst cases that are likely [10]. The absolute absence
Accuracy Study
of interference effects can be taken as ‘‘specificity’’, so
Precision and Trueness specificity = 100% selectivity [13]. The selectivity of a
method can be quantitatively expressed by using the
Robustness
maximum tolerated ratio (TRmax) [14] (i.e. the concen-
Uncertainty and Accuracy Profiles tration ratio of interference (Cint) to analyte (Ca) leading
to a disturbance (systematic error) on the analytical re-
Figure 1. Scheme for the holistic approach to validation.
sponse that yields a biased estimated analyte concen-

228 http://www.elsevier.com/locate/trac


b
tration, C a , falling outside the confidence interval de- ceutiques (SFSTP) [1], several experimental designs are
rived from the expanded uncertainty U at a given available for CSs and validation standards (VSs). For
probability confidence level): example, a typical experimental design for calibration
consists of preparing duplicate solutions at N concen-
Cint tration levels and replicating over three days or three
TRmax ¼
Ca ð3Þ conditions: 3 · N · 2 [8]. VSs are prepared in the matrix
b
C a 62 ½Ca À U; Ca þ UŠ with maybe a different experimental design p · m · n (p
conditions, m levels and n repetitions), as we will con-
In this way, selectivity is supported by uncertainty, sider in Section 2.4. on accuracy study.
and, at least, a crude estimation of the uncertainty is To obtain the best adapted calibration function, sev-
needed for the nominal concentration of analyte (Ca) eral mathematical models can be tested, involving
used in the selectivity study. mathematical transformations as well as weighted
The selectivity, when applying separative analytical regression techniques in which the response variance
methods (e.g., chromatography or capillary electropho- varies as a function of analyte concentration.
resis), is envisaged as the ability of the method to mea-
sure accurately the analyte in presence of all the 2.3.1. Goodness of fit. Suitable regression analysis of the
potential sample components (e.g., placebo formulation, analytical signal (Y) on the analyte concentrations (Z)
synthesis intermediates, excipients, degradation prod- established in the calibration set yields the calibration
ucts, and process impurities) [15], leading to pure, b
curve for the predicted responses ð Y Þ. The simplest
symmetric peaks with suitable resolution [16]. For model is the linear one, very often found in analytical
example, in chromatographic methods, the analyte peak methodology, leading to predicted responses according
in the mixture should be symmetric with a baseline to Equation (4):
resolution of at least 1.5 from the nearest eluting peaks. b
Y ¼ a þ bZ ð4Þ
Non-separative analytical methods (e.g., spectropho-
tometric methods) are susceptible to selectivity problems where a is the intercept and b the slope, with standard
(with the exception of highly selective fluorescence-based deviations sa and sb, respectively. However, as we stated
techniques), but, in some cases, first- and second-deriv- above, non-linear models can be also applied in a
ative techniques may overcome these handicaps [17]. In number of analytical techniques, as given in Equation (5):
electroanalytical methods, selective electrodes or b
Y ¼ f ðZÞ ð5Þ
amperometric sensors are practical examples of specific
f being the non-linear function to be tested.
devices. For voltammetric methods, pulsed differential
Equation (4), or (5), must be checked for goodness of
votammetry and square-wave techniques are suitable to
fit. The correlation coefficient, although commonly
enhance selectivity [18].
used, especially in linear models, is not appropriate
However, sometimes the sensitivity (slope of the cali-
[19], so some more suitable criteria should be consid-
bration function) of the analyte is affected by the sample
ered. A simple way to diagnose the regression model is
matrix, leading to another kind of interference for which
to use residual plots [20–22]. For an adequate model,
the matrix acts as a whole. These effects may be cir-
the residuals are expected to be normally distributed, so
cumvented by using in situ calibration following the
a plot of them on a normal probability graph may be
method of standard additions and we will consider them
useful. Any curvature suggests a lack of fit due to a
in Sections 2.3 and 2.4., on calibration study and the
non-linear effect. A segmented pattern indicates heter-
accuracy study, respectively, because matrix effects may
oscedasticity in data, so weighted regression should be
lead to systematic additive and proportional errors.
used to find the straight line for calibration [23]. In the
latter case, the non-homogeneity of variances can also
2.3. Calibration study
be checked by using the Cochran criterion [24], when
The response function or calibration curve of an ana-
the number of observations is the same for all con-
lytical method is, within the range, a monotonic rela-
centration levels. Another advantage of using calibra-
tionship between the analytical signal (response) and the
tion designs with repetition is the possibility of
concentration of analyte [4]. Response function can be
estimating a pooled sum of squares due to pure errors,
linear, but non-linear models, such as in enzyme-linked
SSPE. The best way to test the goodness of fit is by
immunosorbent assay (ELISA) or inductively coupled
comparing the variance of the lack of fit against the
plasma (ICP) techniques are also observed.
pure-error variance [22].
The response function is obtained using calibration
The residual sum of squares of the model
standards (CSs) prepared in absence of matrix sample
and relating the response and the concentration. X
N
According to the harmonized procedure published by the SSR ¼ b
ð Y i À Y i Þ2 ð6Þ
´
Societe Francaise des Sciences et Techniques Pharma-
¸ i¼1

http://www.elsevier.com/locate/trac 229


can be decomposed into the sum of squares corre- b
linear and without intercept, Y i ¼ bZ i , so RF i ¼ Y ii ’ b.
Z
sponding to pure error (SSPE) and the sum of squares At high concentrations, negative deviation from linear-
corresponding to the lack of fit (SSLOF), hence: ity is expected. Two parallel horizontal lines are drawn in
the graph at 0.95 and 1.05 times the average value of
SSLOF ¼ SSR À SSPE the response factors (very close to b, as indicated above)
ð7Þ
mLOF ¼ mR À mPE in a fashion similar to the action limits of control charts.
The linear range of responses corresponds to the analyte
mPE and mR being the degrees of freedom for estimating concentrations from the point intersecting the line
the sum of squares of pure error and residuals, respec- y = 1.05b up to the point that intersects the line
tively. y = 0.95b. It is possible that no intersections are found,
The pure-error variance is SSPE/mPE, and the variance and, in this case, the linear range applies to the full
of the lack of fit is SSLOF/mLOF. In order to estimate the range being studied, if the minimum concentration level
adequacy of the model, the Fisher F-test is applied: is higher than the limit of detection (LOD) in case of trace
analysis.
ðSSR À SSPE Þ=ðmR À mPE Þ In routine analysis, linear ranges are established in
F¼ ð8Þ
SSPE =mPE assay methods as 80–120% of the analyte level. For
impurity tests, the lower linearity limit is the limit of
The calibration model is considered suitable if F is less quantitation (LOQ), so the range of linearity begins with
than the one-tailed tabulated value Ftab(mR À mPE,mPE,P) the LOQ and finishes about the 150% of the target level
at a P selected confidence level. for the analyte, according to USP/ICH and IUPAC
It is possible, from this proof, to devise a connection guidelines [13].
with correlation coefficient r by remembering that 1 À r2
accounts for the ratio of the residual sum of squares to 2.3.3. Sensitivity, detection limit and quantitation
the total sum of squares of the deviation about the mean, limit. Sensitivity is the change in the analytical re-
´
as Gonzalez et al. pointed out [25]. sponse divided by the corresponding change in analyte
Always within the realm of linear models, and taking concentration; i.e. at a given value of analyte concen-
into account the term ‘‘linearity’’ in the context of lin- tration Z0:
earity of the response function, some authors consider
two features: in-line and on-line linearity [26]. In-line dY
Sensitivity ¼ ð9Þ
linearity refers to the linearity of the model assessed by dZ Z0
the goodness of fit (absence of curvature), and on-line
linearity refers to the dispersion of the data around the If the calibration is linear, the sensitivity is just cali-
calibration line and is based on the relative standard bration slope b at every value of analyte concentration.
deviation of the slope, RSDb = sb/b. This value is taken as In addition to sensitivity, there are two parameters
another characteristic parameter of performance and reciprocally derived from sensitivity, much more often
depends on the maximum value accepted for RSDb, so a used for performance characteristics: the limit of detec-
typical threshold may be RSDb 6 5%. Once both ‘‘in- tion (LOD) and the limit of determination or quantitation
line’’ and ‘‘on-line’’ linearity are assessed, a test for an (LOQ).
intercept significantly different from zero is generally LOD is the lowest concentration of analyte that can be
performed by applying the Student t-test [27]. detected and reliably distinguished from zero (or the
Once the calibration curve is obtained, an inverse noise level of the system), but not necessarily quantified;
prediction equation is then built to predict the actual the concentration at which a measured value is larger
concentrations of the VSs by considering corrections than the uncertainty associated with it. LOD can be
terms, if needed, owing to the presence of constant and expressed in response units (YLOD) and is taken typically
proportional bias. as three times the noise level for techniques with con-
tinuous recording (e.g., chromatography). Otherwise, it
2.3.2. Linear range. The procedure described by Huber is commonly estimated by using the expression [29]:
[28] is quite efficient at giving the proper linear dynamic
Y LOD ¼ Y blank þ 3sblank ð10Þ
range of analyte from the calibration data. It consists of
evaluating so-called response factors RFi obtained by where Yblank and sblank are the average value of the blank
dividing the signal responses by their respective analyte signal and its corresponding standard deviation,
concentrations. A graph is plotted with the response respectively, obtained by measuring at least a minimum
factors on the y-axis and the corresponding concentra- of 10 independent sample blanks. Alternatively, when
tions on the x-axis. The line obtained should be of near- sample blank cannot produce any response (i.e. vol-
zero slope (horizontal) over the concentration range. tammetry), 10 independent sample blanks fortified at the
This behavior is supported assuming that the model is lowest acceptable concentration of the analyte are



measured and then, YLOD = 3s, s being the standard bias due to matrix effects is the standard-addition
deviation of the set of measurements. method (SAM) and the Youden plot [2,33–35].
Nevertheless, LODs expressed in signal units are awful Consider the external standard calibration curve, ob-
to handle. It is more advisable to use LODs in analyte- tained by plotting the signal or analytical response of
concentration units. Thus YLOD values are converted to different standard solutions of the analyte. Let us assume
ZLOD by using the calibration function: that the standard calibration relationship is linear within
a given concentration range of analyte, so the analytical
Y LOD À a
ZLOD ¼ ð11Þ response follows Equation (4).
b Consider now the application of the analytical proce-
Then, the final LOD value, considering zero intercept, dure to a dissolved test portion of a unknown sample
gives: within the linear working range. Assuming that the
sample matrix does not contribute to the signal as an
3sblank
ZLOD ¼ ð12Þ interfering agent [36] and that there is no interaction
b between the analyte and the matrix, the analytical
If the errors associated with the calibration line are response can be now modeled as:
taken into account [30], another expression can be
b
Y ¼ A þ BZ ð15Þ
applied [31]:
2 1=2 where A and B are sample constants. A is a constant that
2tðm; PÞ½s2 þ s2 þ ða=bÞ s2 Š
blank a b
ZLOD ¼ ð13Þ does not change when the concentration of the analyte
b
and/or the sample change [37]. It is called the ‘‘true
LODs have to be determined only for impurity methods sample blank’’ [38] and can be evaluated from the
but not for assay methods. YoudenÕs sample plot [39–41]. B is the fundamental
LOQ is the lowest concentration of analyte that can be term that justifies the analytical procedure, and it is di-
determined quantitatively with an acceptable level of rectly related to the analytical sensitivity [42]. If both
precision [31]. The procedure for evaluating LOQs is constant and proportional bias are absent, then A = a
equivalent to that of LODs, by measuring at least 10 and B = b. In order to assess the absence of proportional
independent sample blanks and using the factor 10 bias, a homogeneous bulk spiked sample from a matrix
instead of 3 for calculations: that contains the analyte is used. The analyte (here,
Y LOQ ¼ Y blank þ 10sblank ð14Þ surrogate) has to be spiked at several concentration
levels in order to cover the concentration range of the
To express LOQ in concentration units, relationships method scope. Here the SAM can be suitably used to
equivalent to Equations (12) and (13) can be applied estimate the recovery of spiked samples [2,33–35], so,
by changing LOD into LOQ. The reason for the factor for a spiked sample, Equation (15) may be rewritten as:
10 comes from IUPAC considerations [31], assuming a
relative precision of about 10% in the signal. However, b
Y ¼ A þ BðCnative þ Cspike Þ ¼ A þ BCnative þ BCspike
in order to obtain an LOQ more consistent with the ¼ aSAM þ bSAM Cspike ð16Þ
definition, it is advisable to make a prior estimation of
the RSD of the response against the analyte concen- where Cnative is the concentration of the analyte in the
tration (near to the unknown LOQ). Thus, a series of unspiked sample, Cspike the concentration of the spiked
blanks are spiked at several analyte concentrations and analyte, and aSAM and bSAM are the intercept and the
measured in triplicate. For every addition, the %RSD is slope of the SAM calibration straight line.
calculated. From the plot of %RSD versus the spiked From Equation (16), we get:
analyte concentration, the amount that corresponds to bSAM ¼ B
a previously defined precision RSD is interpolated and ð17Þ
aSAM ¼ A þ bSAM Cnative
taken as the ZLOQ [32]. As indicated above, LOQs are
immaterial for assay methods, but, for impurity tests If we try to estimate the analyte concentration of a
and trace analysis, the lower linearity limit is always spiked sample by using the external calibration line
the LOQ [13]. (Equation (4)), we obtain an estimation of the total ob-
served analyte concentration:
2.3.4. Checking proportional and constant bias derived from b
b Y À a ðaSAM À aÞ þ bSAM Cspike
matrix effects. As stated above, the use of external cal- C obs ¼ ¼ ð18Þ
ibration enormously simplifies the protocol because cal- b b
ibration standards are prepared as simple solutions of the For the unspiked sample (Cspike = 0), an estimation of
analyte. However, the effects of possible matrix effects the native analyte concentration is obtained:
coming from the sample material must be checked. A b aSAM À a
C native ¼ ð19Þ
very useful tool for testing constant and proportional b



According to Equations (18) and (19), the spiked Table 1. Acceptable recovery percentages depending on the
concentration of analyte is estimated from the external analyte level
calibration as:
Analyte (%)
% Analyte Unit Recovery
b b b bSAM fraction range (%)
%
C spike ¼ C obs À C native ¼ Cspike ð20Þ
b
100 1 100% 98–102
The relationship established by Equation (20) is of the 10 10À1 10% 98–102
utmost importance because it leads to a measure of the 1 10À2 1% 97–103
overall consensus recovery: 0.1 10À3 0.1% 95–105
0.01 10À4 100 ppm 90–107
b
C spike bSAM 0.001 10À5 10 ppm 80–110
R¼ ¼ ð21Þ 0.0001 10À6 1 ppm 80–110
Cspike b
0.00001 10À7 100 ppb 80–110
The absence of proportional bias corresponds to 0.000001 10À8 10 ppb 60–115
0.0000001 10À9 1 ppb 40–120
bSAM = b, or, in terms of recovery, R = 1. This must be
checked for statistical significance [43]:
jR À 1j
t¼ ð22Þ
uðRÞ Y ¼ A þ bYouden wsample ð24Þ
with the uncertainty given by: The intercept of the plot is an estimation of the total
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Youden blank (TYB), which is the sum of the system
u2 ðbSAM Þ b2 u2 ðbÞ blank (SB) corresponding to the intercept of the
uðRÞ ¼ þ SAM 4 ð23Þ
b2 b standard calibration (a) and the Youden blank (YB)
According to the LGC/VAM protocol [44], if the associated with the constant bias in the method
degrees of freedom associated with the uncertainty of [34,46]. Thus, we can equate TYB = A, SB = a and
consensus recovery are known, t is compared with the YB = A À a. The constant bias in the method is defined
two-tailed tabulated value, ttab(m,P) for the appropriate as [34]:
number of degrees of freedom at P% confidence. If YB A À a
t 6 ttab, the consensus recovery is not significantly dif- hc ¼ ¼ ð25Þ
b b
ferent from 1. Alternatively, instead of ttab, coverage
The uncertainty of the constant bias can be obtained
factor k may be used for the comparison. Typical values
by the law of variance propagation:
are k = 2 or k = 3 for 95% or 99% confidence, respec- sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
tively [45], so u2 ðAÞ u2 ðaÞ ðA À aÞ2 u2 ðbÞ 2ðA À aÞ
if jRÀ1j 6 k, the recovery is not significantly different uðhc Þ ¼ þ 2 þ þ covða; bÞ
uðRÞ b2 b b4 b3
from 1; and,
if jRÀ1j k, the recovery is significantly different from 1 ð26Þ
uðRÞ
and the analytical result must be corrected by R. The uncertainties, u2(A), u2(a) and u2(b), are obtained
Recovery is sometimes considered a separate valida- from the statistical parameters of the straight line fits:
tion parameter, but, in any case, it should always be s2(A) from the YoudenÕs plot; and, s2(a) and s2(b) from
established as a part of method validation [13]. Apart the external calibration plot. Also cov(a,b) is computed
from the statistical significance given above, there are from the external calibration straight line.
published acceptable recovery percentages as a function Once uncertainty u(hc) is evaluated, the constant bias
of the analyte concentration [28], as shown in Table 1. in the method is tested for significance in a way very
In any case, the relative uncertainty for proportional similar to recovery:
bias due to matrix effects, according to the SAM, is taken if uðhccjÞ 6 k, the constant bias is not significantly differ-
jh

as uðRÞ.
R ent from 0; and,
As mentioned above, in the presence of sample matrix, if uðhccjÞ k, the constant bias is significantly different
jh

the relationship between the analytical response and the from 0 and the analytical result should be corrected
analyte concentration is given by Equation (15). The by hc
independent term ‘‘A’’ is the true sample blank because As Maroto et al. [47] pointed out, even if the analytical
it is determined when both the native analyte and the procedure is free from constant bias, its uncertainty must
matrix are present. The SAM calibration, indicated by be included in the overall uncertainty budget for future
Equation (22), includes this term within the intercept determinations. The same applies to the absence of
(aSAM = A + bSAMCnative). The YoudenÕs plot [39–41] proportional bias. Thus, consider Zfound, the analyte
consists of plotting the instrumental response (Y) against concentration obtained by applying the analytical pro-
the amount of sample (the weight or volume of sample cedure to a sample by using the external calibration
test portion to be dissolved up to the assay volume): function. If, in the matrix-effect study, there are both



proportional bias (recovery R significantly different from reliability during normal usage. Robustness tests can be
1) and constant bias (offset hc significantly different from considered to be intra-laboratory simulations of inter-
0), then the corrected estimated concentration value for laboratory studies, if the alterations introduced are
the analyte will be: suitably selected. Usually, robustness tests that yield
significant effects for the measurand lead to further
Zfound À hc
Z¼ ð27Þ optimization of the method, so uncertainty evaluation
R should be performed only after the method has been
Another way to obtain the corrected concentration shown to be robust.
directly is to use the corrected calibration equation for In the case of inter-laboratory validation, the expres-
the measured response Y: sion for uncertainty is:
YÀA u2 ðZÞ ¼ S2 þ u2 ðdÞ
R ð30Þ
Z¼ ð28Þ
bSAM Now SR is the inter-laboratory reproducibility, and
The values of A and bSAM are previously established for hence it is not needed any robustness study.
every kind of matrix subjected to validation and estab- Because inter-laboratory measurements can be done
lished in the method scope and applicability. only when inter-laboratory exercises are available, we
Equation (28) is the inverse prediction equation to be will focus on the intra-laboratory estimation of uncer-
used for estimating the actual concentrations of VSs. tainty.
Note that once it has been demonstrated that both The estimation of bias and reproducibility is performed
constant and proportional bias are absent for the using suitable VSs, prepared in the same matrix as that
matrices considered for validation, CSs could be taken as expected for future samples. Certified or internal refer-
VSs without problem. ence materials represent the best way to obtain a VS, but
spiked samples can be considered as a suitable alterna-
2.4. Accuracy study tive [8]. In the case of pharmaceutical formulations (or
As stated above, method validation scrutinizes the other manufactured products) where a ‘‘placebo’’ is
accuracy of results by considering both systematic and available, the bias or precision study should be carried
random errors. Accuracy is therefore studied as an entity out using spiked placebos. But, when a placebo is not
with two components – trueness and precision – but available, selected stable samples fortified to a suitable
considered as a global entity, the uncertainty [48], from level of the analyte may be prepared. VSs must be stable,
which the bETI will be estimated as well as the accuracy homogeneous and as similar as possible to the future
profiles once the acceptability limits have been estab- samples to be analyzed, and they represent, in the vali-
lished. dation phase, the future samples that the analytical
We are interested in estimating the accuracy profile procedure will have to quantify. Each VS must be pre-
from the uncertainty measurement of the analytical pared and treated independently as a future sample. This
assay from validation data according the LGC/VAM independence is essential for a good estimation of the
protocol [44] and the ISO/DTS 21748 guide [49]. The between-series variance. Indeed, the analytical proce-
basic model for the uncertainty of measurand Z is given dure is not developed to quantify routinely with the same
by three terms for intra-laboratory measurements: operator and on the same equipment a single sample
unknown on one day but a very large number of
u2 ðZÞ ¼ S2 þ u2 ðdÞ þ u2 ðZÞ
R rob ð29Þ
samples through time, thus often implying several
where SR is the intra-laboratory-reproducibility standard operators and several equipments.
deviation (intermediate precision), u(d) is the uncer-
tainty associated with the bias or trueness of the proce- 2.4.1. Intermediate precision and trueness studies. Both
dure, and urob is the uncertainty coming from a intermediate precision and trueness studies can be per-
robustness exercise. formed using the prediction of actual concentrations
As both precision and trueness are assessed within a from the VSs selected for the analytical assay. Following
single laboratory, the uncertainty due to laboratory the golden rules of validation, the analytical procedure
transfer should be taken into account. This can be ob- should be validated separately for each kind of matrix
tained from the robustness study, which considers considered, covering the full range of analyte concen-
changes in the variables of the analytical procedure trations.
(called factors) expected in a transfer between laborato- Accordingly, a suitable way to perform the inter-
ries. According to the International Conference on Har- mediate precision study is to consider a single sample
monization (ICH Q2A document) [50], the robustness of matrix and a range of analyte concentrations. It is
an analytical procedure is a measure of its capacity to advisable for there to be at least three concentration
remain unaffected by small, but deliberate, variations in levels m (low, medium and high) covering the dynamic
method parameters, and provides an indication of its working range, with a number of n replicates at each



concentration. The ICH Q2B document recommends three ation, RSD (either for repeatability or reproducibility),
replicates [51] and the FDA document on bioanalytical calculated from the analytical data to the Horwitz value:
validation considers five replications [52], so 3–5 repli- RSD
cations are advisable. Calculations of intermediate preci- Horrat ¼ ð37Þ
PRSDR
sion must be carried out on results instead of responses.
Considering the different conditions p (here, the days) Apart from HorwitzÕs parameters, expected values of
chosen as the main source of variation, an analysis of RSD according the AOAC Peer Verified Methods program
variance (ANOVA) is then performed for each concen- are also considered [59]. These two approaches, as a
tration. Accordingly, for each concentration level, m, we function of the analyte concentration, are presented in
consider the results of the analysis, according to inverse- Table 2. Some practical requirements concerned with
prediction Equation (28), zij with two indices: i (from 1 to inter-laboratory studies are [13]:
p) corresponding to the different days and j (from 1 to n) RSDr = 0.5–0.6 times PRSDR
accounting for the repetitions. From the ANOVA, we can RSDR = 0.5–2 times PRSDR
easily obtain [53,54] estimations of within-condition For intra-laboratory validations, a quick rule is to di-
variance ðS 2 Þ and between-condition variance ðS 2 Þ. The
W B
vide the interval by 2 [60], leading to:
within-condition, also known as repeatability, variance RSDr = 0.2–0.3 times PRSDR
ðS 2 Þ is given by:
r
RSDR = 0.2–1 times PRSDR
In the bias calculation, the value is taken as the final
z
PP
p n
ðzij À i Þ2
z result for Z corresponding to the VS of estimated ‘‘true’’
i¼1 j¼1 concentration T, so the average bias for the VS is ob-
S2 ¼ S2 ¼
W r ð31Þ
pðn À 1Þ tained from the elemental bias dij = zij À T according to:
1 XX 1 XX
with p n p n
d¼ dij ¼ zij À T ¼ À T
z
P
n pn i¼1 j¼1 pn i¼1 j¼1
zij
i ¼
z
j¼1
ð32Þ ¼ZÀT ð38Þ
n
The bias uncertainty can be estimated from the same
The between-condition variance is estimated accord- ANOVA design, according to ISO/DTD 21748 guide
ing to: [49], as:
P
p c
S2 ð1 À c þ nÞ
ði À Þ2
z z u2 ðdÞ ¼ R
ð39Þ
S2 p
S2 ¼ i¼1
B À r
ð33Þ
pÀ1 n
with
with
S2
r
PP
p n c¼ ð40Þ
zij S2
R
¼ i¼1
z
j¼1
ð34Þ Accordingly, the only term we need to estimate to
pn
evaluate u(Z) from Equation (29) is the robustness
The intra laboratory reproducibility or intermediate uncertainty.
precision can be taken as:

S2 ¼ S2 þ S2
R r B ð35Þ

From these data, the corresponding relative standard Table 2. Acceptable RSD percentages obtained from the Horwitz
function and from the AOAC Peer Verified Methods (PVM)
deviations, RSDr and RSDR, are calculated. These values program on the analyte level
can be compared with the expected values issued from
the Horwitz equation and the ‘‘Horrat’’ [55,56]. Horwitz Analyte (%) Analyte Unit Horwitz AOAC PVM
fraction %RSD %RSD
[57] devised an expression to predict the expected value
of the relative standard deviation for inter-laboratory 100 1 100% 2 1.3
reproducibility (PRSDR) according to: 10 10À1 10% 2.8 1.8
1 10À2 1% 4 2.7
PRSDR ¼ 2ð1À0:5 log CÞ ð36Þ 0.1 10À3 0.1% 5.7 3.7
0.01 10À4 100 ppm 8 5.3
where C is the analyte concentration in decimal fraction 0.001 10À5 10 ppm 11.3 7.3
units. The Horwitz value is now widely used as a 0.0001 10À6 1 ppm 16 11
benchmark [58] for the performance of analytical 0.00001 10À7 100 ppb 22.6 15
methods via a measure called the ‘‘Horrat’’, which is 0.000001 10À8 10 ppb 32 21
0.0000001 10À9 1 ppb 45.3 30
defined as the ratio of the actual relative standard devi-



2.4.2. Robustness study. Robustness [61,62], consid- experiments. The corresponding matrix design is illus-
ered in the sense of internal validation, deals with the trated in Table 3. The eight runs are split into two
effect of experimental variables, called factors, inherent groups of four runs on the basis of levels +1 or À1. The
in the analytical procedure (e.g., temperature, mobile- effect of every factor xk is estimated as the difference of
phase composition, detection wavelength, and pH), on the mean result obtained at the level +1 from that ob-
the analytical result. A robustness study examines the tained at the level À1.
alteration of these factors, as expected in a transfer be- 0 ! ! 1
tween laboratories, so is of the utmost importance in the 1@ X N XN
A
Dðxk Þ ¼ Zi À Zi ð41Þ
uncertainty budget. In experiments to study the main 4 i¼1 i¼1
ðxk ¼þ1Þ ðxk ¼À1Þ
effects of factors, screening designs are used. Screening
designs are two-level saturated fractional factorial Once effects D(xk) have been estimated, to determine
designs centered on the nominal analytical conditions whether variations have a significant effect on the result,
[63]. Plackett and Burmann [64] developed such designs a significance t-test is used [68]:
for studying f factors in N = f + 1 experiments, where N pffiffiffi
2jDðxk Þj
is any multiple of 4 less than 100 (except for 92) [65]. tðxk Þ ¼ ð42Þ
Plackett-Burmann designs are very useful tools for a SR
robustness study of analytical procedures. However, The t-value is compared with the 95%-confidence level
these designs cannot deal with factor interactions, so two-tailed tabulated value with the degrees of freedom
they are suitable only when the interactions are negli- coming from the precision study for each concentration:
gible or when considering a key set of dominant factors m = pn À 1. If t(xk) 6 ttab, then the procedure is robust
[25]. against the factor xk. In this case, the uncertainty on
The strategy for carrying out a robustness study is measurand Z coming from factor xk, u(Z(xk)) is evaluated
based on a landmark procedure suggested by Youden from [44]:
[66,67]: tcrit SR dreal ðxk Þ
(1) identify the influential factors; uðZðxk ÞÞ ¼ pffiffiffi ð43Þ
1:96 2 dtest ðxk Þ
(2) for each factor, define the nominal and the extreme
values expected in routine work and encode them as Here dreal is the change in the factor level that would
follows: nominal value = 0, high value = +1 and be expected when the method is operating routinely, and
low value = À 1; dtest is the change in the factor level specified in the
(3) arrange the experimental design by using a two- robustness study. Once the contributions of the influ-
level 27À4 fractional Plackett-Burmann matrix; and, ential factors have been estimated, the relative uncer-
(4) perform the experiments in random order on a con- tainty of the robustness study is calculated as:
trol sample with analyte concentration halfway in vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
uP 2
the concentration range of the method scope. u u ðZðxk ÞÞ
tk
According to the definition of robustness, the interval RSDrob ¼ ð44Þ
Z2
under investigation is very short (À1, +1; e.g., pH 3.8–
4.2). Under these conditions, it must be stressed that no Z is the average value of the eight results obtained in the
quadratic effect is generally observed, so a linear model robustness study. By taking this value for any future
can be used. It is one of fundamental differences between concentration, Z, the uncertainty of robustness is urob(Z) =
the robustness study and the optimization study, in Z RSDrob. This leads to the final uncertainty budget
which the interval under investigation is wider. according to Equation (29). The expanded uncertainty is
Youden selected a 27À4 Plackett-Burmann design be- obtained as the product of the standard uncertainty u(Z)
cause it enabled the study of up to seven factors in eight and the coverage factor k selected as the b quantile of the

Table 3. Arrangement of factor levels for a 27À4 Plackett-Burmann design

Runs Factors xk (k = 1 to f) Response
N X1 X2 X3 X4 X5 X6 X7 Zi (i = 1 to N)
1 +1 +1 +1 +1 +1 +1 +1 Z1
2 +1 +1 À1 +1 À1 À1 À1 Z2
3 +1 À1 +1 À1 +1 À1 À1 Z3
4 +1 À1 À1 À1 À1 +1 +1 Z4
5 À1 +1 +1 À1 À1 +1 À1 Z5
6 À1 +1 À1 À1 +1 À1 +1 Z6
7 À1 À1 +1 +1 À1 À1 +1 Z7
8 À1 À1 À1 +1 +1 +1 À1 Z8



Student t distribution, so the final expression for the con- racy profile can be constructed, according to Gonzalez ´
fidence interval of the analytical result is and Herrador [69]: for each concentration level, the
Z Æ kuðZÞ ¼ Z Æ UðZÞ ð45Þ bETI is calculated from Equation (47). Upper and lower
tolerance-interval limits are then connected by straight
lines to interpolate the behavior of the limits between the
2.4.3. Study of accuracy profiles. Considering that the levels studied. The quantification limits are located at the
concentrations of VSs are taken as reference values T, intersections between the interpolating lines and the
after analyzing them, the interval for the bias of the acceptance limits. If we deal with three levels – low (Tl),
result is given, according to Equation (38), as: medium (Tm) and high (Th) – three bETIs are computed
ðZ Æ UðZÞÞ À T ¼ ðZ À TÞ Æ UðZÞ ¼ d Æ UðZÞ ð46Þ and expressed as percentage values – 100 dl ÆUlðZ l Þ ; Z
100 dm ÆUmðZ m Þ and 100 dh ÆUhðZ h Þ. The points corresponding
Z Z
Remembering Equation (39) and condition (2), the
to the upper limits have coordinates ðT l ; 100 dl þUlðZ l ÞÞ;
corresponding bETI is constructed as: Z
ðT m ; 100 dm þUmðZ m ÞÞ and ðT h ; 100 dh þUhðZ h ÞÞ, and they are
Z Z
jd Æ UðZÞj k ð47Þ connected by linear segments. The same procedure is
When the bETI is evaluated for a number of concen- done for the lower limits ðT l ; 100 dl ÀUlðZ l ÞÞ;
Z
dm ÀU ðZ m Þ dh ÀU ðZ h Þ
tration levels (VSs) covering the whole range of analyte ðT m ; 100 Z m Þ and ðT h ; 100 Z h Þ. The intersec-
concentrations specified in the method scope, the accu- tions between these two segmented lines with the

y=λ
y=λ
B
B C

y = -λ
y = -λ
A
A D

Tl Tm Th
Tl Tm Th
Figure 2. Accuracy profile when intersection occurs between med-
ium and low level. LQL = B (B A) and UQL = Th (no intersections Figure 4. Accuracy profile when several intersections take place.
in this zone). LQL = B (B A) and UQL = C (C D).

y=λ
A y=λ

y = -λ
B y=λ

Tl Tm Th
Tl Tm Th
Figure 3. Accuracy profile when two intersections occur, one
between medium and low level and another between medium Figure 5. Accuracy profile when no intersections occur in the
and high level. LQL = A and UQL = B. working range of concentration.



acceptance-limit straight lines y = k and y = À k (in %) [7] P. Hubert, J.J. Nguyen-Huu, B. Boulanger, E. Chapuzet, P. Chiap,
give the aforementioned quantification limits. Some N. Cohen, P.A. Compagnon, W. Dewe, M. Feinberg, M. Lallier,
M. Laurentie, N. Mercier, G. Muzard, C. Nivet, L. Valat, J. Pharm.
typical patterns are shown in Figs. 2–5. The domain of Biomed. Anal. 36 (2004) 579.
valid concentrations for future assays corresponds to the [8] M. Feinberg, B. Boulanger, W. Dewe, P. Hubert, Anal. Bioanal.
quantification limits LQL (lower) and UQL (upper) that Chem. 380 (2004) 502.
can be extracted from accuracy profiles. [9] M. Feinberg, M. Laurentie, Accred. Qual. Assur. 11 (2006) 3.
[10] M. Thompson, S.L.R. Ellison, R. Wood, Pure Appl. Chem. 47
(2002) 835.
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3. Summary P.J. Lewi, J. Smeyers-Verbeke, Handbook of Chemometrics and
Qualimetrics, Part A, Elsevier, Amsterdam, The Netherlands,
In this article, we have presented in detail a holistic 1997.
approach to validate analytical methods including [12] Eurachem, Guide: The Fitness for Purpose of Analytical Methods:
A Laboratory Guide to Method Validation and Related Topics,
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method, considering its applicability, fitness for purpose 23 (2004) 535.
and the given acceptability limits, paying special atten- ´
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´
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because of its importance in method validation. At this ´
[19] A.G. Asuero, A. Sayago, A.G. Gonzalez, CRC Anal. Chem. 36
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´ ´ ˜
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[53] International Organization for Standardization (ISO), ISO-5725-2, ´
A. Gustavo Gonzalez received his Ph.D. in Chemistry from the Uni-
Accuracy (Trueness and Precision) of Measurement Methods and versity of Seville, Spain, in 1986. His research has focused on several
Results - Part 2: Basic Method for the Determination of Repeat- lines: solvent effects on dissociation and drug solubility; chemometric
ability and Reproducibility of a Standard Measurement Method, pattern recognition covering methodological developments (including
ISO, Geneva, Switzerland, 1994. software) and its application to the authentication of foods and bev-
[54] T. Luping, B. Schouenborg, Methodology of Inter-comparison erages; experimental design and optimization of the analytical process;
Tests and Statistical Analysis of Test Results. Nordtest Project No. and, quality assurance, validation of analytical methods and estimation
1483-99, SP Swedish National Testing and Research Institute, SP of the measurement uncertainty. He has published over 120 scientific
˚
report 2000:35, Boras, Sweden, 2000. papers on these subjects.
[55] M. Thompson, The Amazing Horwitz Function, AMC Technical
Brief No. 17 Royal Society of Chemistry, Cambridge, UK, July ´
M. Angeles Herrador obtained her Ph.D. in Pharmacy from the
2004. University of Seville in 1985. Her research interests comprise the
[56] R. Wood, Trends Anal. Chem. 18 (1999) 624. development of methodologies for pharmaceutical and biomedical
[57] W. Horwitz, Anal. Chem. 54 (1982) 67A. analysis, and quality assurance for drug assays. In 2002, she obtained
[58] Codex Alimentarius Commission, Codex Committee on Methods of the title of Pharmacist Specialist in Analysis and Control of Drugs from
Analysis and Sampling, CX/MAS 01/4, Criteria for evaluating ´
the Ministerio de Educacion, Cultura y Deporte in Spain. She has
acceptable methods of analysis for Codex purposes, Agenda Item published about 70 scientific papers on these topics.


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